4:35pm - 5:50pm on Mondays and Wednesdays in 427 Kaprielian Hall (KAP).
We will assume familiarity the material from Math 540a and Math 535a.
This is a second semester graduate level course in algebraic topology. We will cover various advanced topics, sampling many of the highlights from the 20th century while building the foundations for modern research in topology.
| $\#$ | Date | Material | References | Problem set |
|---|---|---|---|---|
| 1 | Monday 1/12/26 | Topological manifolds and Poincaré dualities. Geometric argument for surfaces using dual cell divisions. Local and global orientations. $R$-orientations. The orientable double cover of a topological manifold. Fundamental classes. | [H1] §3.3 | |
| 2 | Wednesday 1/14/26 | A few remarks on the existence of triangulations and cell structures on topological or smooth manifolds. More on orientations and fundamental classes. Begin proof on the existence of fundamental classes over $R$ for closed $R$-orientable manifolds. | [H1] §3.3 | |
| Monday 1/19/26: MLK Day - no class | ||||
| 3 | Friday 1/23/26 | More on the existence of fundamental classes. Recalling the universal coefficients theorem for homology. Computing the torsion of $H_{n-1}(M)$ for a topological $n$-manifold. | [H1] §3.3 | |
| 4 | Monday 1/26/26 | Noncompact manifolds $M^n$ have vanishing homology in degrees $\geq 2$. Brief review of singular homology and cohomology. Definition of the cap product and proof that it descends to homology. Naturality of the cap product under continuous maps. Statement of Poincaré duality using the cap product. | [H1] §3.3 | |
| 5 | Wednesday 1/28/26 | Cohomology with compact support. Direct limits of abelian groups. Statement of Poincaré duality for noncompact manifolds. | [H1] §3.3 | |
| 6 | Wednesday 2/4/26 | Proof of Poincaré duality. Relationship between cap and cup products, and implications for nondegeneracy of the cup product (at least with field coefficients). | [H1] §3.3 | |
| 7 | Friday 2/6/26 | Some extensions and consequences of Poincaré duality: topological manifolds with boundary decomposed as $A \cup B$, Alexander duality. Odd-dimensional manifolds have vanishing Euler characteristic. Moore spaces and Eilenberg-MacLane spaces. Review of weak equivalences and Whitehead's theorem. Statement of representability theorem for ordinary cohomology. Eilenberg-MacLane spaces as loop spaces. Existence of Moore spaces. | ||
| 8 | Monday 2/9/26 | Proof of the existence of Eilenberg-MacLane spaces and their uniqueness up to (weak) homotopy equivalence. Recollections on cellular approximation, CW approximation, and smooth approximation. Homotopy groups only depend on the skeleton of dimension one greater. Defining the map involved in the representability theorem for ordinary cohomology. | ||
| 9 | Wednesday 2/11/26 | Proof that the representability map for ordinary cohomology is an isomorphism. Generalities on obstruction theory. The Eilenberg-Steenrod axioms. Begin proof that they characterize ordinary cohomology. | ||
| 10 | Friday 2/13/26 | Proof sketch that the Eilenberg-Steenrod axioms characterize (co)homology. Alternative proof that Eilenberg-MacLane spaces represent cohomology via the Eilenberg-Steenrod axioms. The Puppe sequence. Adjunction between (reduced) suspension and base loop space. | ||
| Monday 2/16/26: Presidents' Day - no class | ||||
| 11 | Wednesday 2/18/26 | |||
| 12 | Friday 2/20/26 | |||
| 13 | Monday 2/23/26 | |||
| 14 | Wednesday 2/25/26 | |||
| 15 | Friday 2/27/26 | |||
| 16 | Monday 3/2/26 | |||
| 17 | Wednesday 3/4/26 | |||
| March 9 - March 27: Extended spring break - no class (3 weeks) | ||||
| 18 | Monday 3/30/26 | |||
| 19 | Wednesday 4/1/26 | |||
| 20 | Monday 4/6/26 | |||
| 21 | Wednesday 4/8/26 | |||
| 22 | Monday 4/13/26 | |||
| 23 | Wednesday 4/15/26 | |||
| 24 | Monday 4/20/26 | |||
| 25 | Wednesday 4/22/26 | |||
| 26 | Monday 4/27/26 | |||
| 27 | Wednesday 4/29/26 |